Abstract

An asymptotic analysis is performed for thin anisotropic elastic plate clamped along its lateral side and also supported at a small area θh\theta_{h} of one base with diameter of the same order as the plate thickness h≪1h\ll 1. A three-dimensional boundary layer in the vicinity of the support θh\theta_{h} is involved into the asymptotic form which is justified by means of the previously derived weighted inequality of Korn's type provides an error estimate with the bound ch1/2∣ln⁡h∣ch^{1/2}\left\vert \ln h \right\vert. Ignoring this boundary layer effect reduces the precision order down to ∣ln⁡h∣−1/2\left\vert \ln h\right\vert ^{-1/2}. A two-dimensional variational-asymptotic model of the plate is proposed within the theory of self-adjoint extensions of differential operators. The only characteristics of the boundary layer, namely the elastic logarithmic potential matrix of size 4×4,4\times4, is involved into the model which however keeps the precision order h1/2∣ln⁡h∣h^{1/2}\left\vert \ln h\right\vert in certain norms. Several formulations and applications of the model are discussed.

Author information

Contact details are reproduced from the original publication and may be historical.

Giuseppe Buttazzo

Dip. di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

buttazzo@dm.unipi.it

Sergei A. Nazarov

Mathematics and Mechanics Faculty, St. Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg 199034, Russia
and: Peter the Great Saint-Petersburg State Polytechnical University, Laboratory “Mechanics of New Nano-materials”, Polytechnicheskaya ul., 29, St. Petersburg, 195251, Russia
and: Institute of Problems of Mechanical Engineering RAS, Laboratory “Mathematical Methods in Mechanics of Materials”, V.O., Bolshoj pr., 61, St. Petersburg, 199178, Russia

s.nazarov@spbu.ru

Keywords

  • Kirchhoff plate
  • small support zone
  • asymptotic analysis
  • self-adjoint extensions
  • variational model

Mathematics Subject Classification

74K20, 74B05

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Suggested citation

G. Buttazzo, G. Cardone, S. A. Nazarov. “Thin Elastic Plates Supported over Small Areas. II: Variational-Asymptotic Models.” Journal of Convex Analysis 24 (2017), No. 3, 819–855. https://doi.org/10.68381/jca24050

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